98 problems

Let $\mathsf{M}$ and $\mathsf{N}$ be matroids on the same ground set, and suppose there is a rank-preserving weak map $\mathsf{N}\to\mathsf{M}$. Let…

Let $P$ be a join-semilattice, and let $\operatorname{Id} P$ denote its lattice of ideals. An ordered set is order-scattered if it does not contain a copy of the order of the ratio…

For $3\leq k\leq 6$, let $\mathcal{X}_{n,k,g}$ be the class of graphs considered in the source and define … where $g$ is fixed sufficiently large and $\tau(G)$ is the number of spa…

Finiteness conjecture. If $\Gamma$ is an irreducible lattice in a higher rank Lie group (or in a nontrivial product of locally compact groups) $G$, there are finitely many Gromov h…

Let $\Gamma$ be a lattice in a higher-rank Lie group, and let its absolute rank gradient be the infimum of $(d(H)-1)/[\Gamma:H]$ over finite-index subgroups $H\leq\Gamma$, where…

Let $G$ be the product of $n\geq 2$ copies of $\operatorname{SL}(2,\mathbb{R})$. Let $U_1$ and $U_2$ be the products of, respectively, upper- and lower-unipotent one-parameter subg…

Let … where each $K_i$ is a local field, each $G_i$ is an absolutely almost simple $K_i$-group, and … Let $\rho(\Gamma)$ denote the abscissa of convergence of the representation ze…

Shioda's conjecture. If $\mathfrak{p}$ is a supersingular prime, then the two lattices $T_Z$ and $T_{\mathfrak{p}}$ are similar.

Bounded-generation conjecture. If $\Gamma$ is any noncocompact lattice in either

Let $\Theta_{\Lambda}(x)$ be the theta series of a $d$-dimensional lattice, and let $\mathcal{P}_n$ denote the class of generating functions whose $n$th roots have integral coeffic…

Let $n\geqslant 3$, let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice, and let $\Pi$ be a sail generated by $\Lambda$. A sail's facets and edge stars are its facets an…

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, m…

Let $v$ be the invariant assigned to the lattices indexed by $1,\ldots,24$, with the indices $19$ and $20$ referring to the corresponding entries in the preceding table. Conjecture…

Let $H$ be a connected semi-simple Lie group without almost simple factors of type $\mathrm{A}_1$ and without compact factors. For sufficiently large $x$, consider maximal irreduci…

Self-dual relaxed lattice conjecture. In every dimension, the largest possible value of $r_1$ among self-dual relaxed lattices equals the smallest value of $r$ possible in Theorem…

Multiplicability conjecture for finitary upho lattices. Every finitary upho lattice, and in particular every finite-type $\mathbb{N}$-graded upho lattice, is multiplicable.

Lattice-realization conjecture. There exists a finite graded lattice $L_\tau$ of rank $n$ such that

For a finite undirected graph $G$, let $\mathsf{B}$ be the graphical pointed building set of $G$, and let $\operatorname{Bicl}(\mathsf{B})$ denote the subposet of biclosed ornament…

Let $C$ be a PULB-optimal code, and let $D$ be its set of universal minima. Assume that $D$ is in general position, meaning that $D$ is not contained in a hyperplane. Universal pol…

Naor's lattice quotient conjecture. There exists a constant $c>0$ such that for every $r\in\mathbb{N}$ and every lattice $\Lambda\subset\mathbb{R}^r$,

Let $G$ be an $S$-arithmetic group in arbitrary characteristic, and let the filling functions and rank be understood as in the cited Lie-group theorem. The S-arithmetic polynomial…

Let $G$ be a non-compact semisimple group over a local field $F$ with finite center and rank $r$. The non-uniform lattice extension conjecture. The full statement of Gromov's $L^p$…

Let $G$ be an $S$-semisimple group of rank $r$ and let $\Gamma<u001b[0mG$ be an irreducible lattice, meaning that its projection to each simple factor of $G$ is dense. Let $V$ be a…

Let $P$ and $Q$ be posets. Write $P\times Q$ for their Cartesian product, ordered componentwise, and let $\dim(P)$ denote the least integer $n$ such that $P$ is isomorphic to a sub…

Let a lattice be a discrete subgroup of Euclidean space, and let its shortest nonzero integer vector length be the relevant lattice parameter. Lattice shortest-vector hardness conj…